I study Metric spaces and I has this problem
Show that sphere $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ is metrically homogeneous. For the other hands, if $\Bbb B=\{x\in \Bbb R^{n+1}; \langle x,x\rangle<1\}$ be a open ball from Euclidean Space $\Bbb R^n$, all isometric g:$\Bbb B\rightarrow\Bbb B$ be such that g(0)=0. So conclude B is not metrically homogeneous.
Does anyone know how can I solve?
For proving that $\mathbb{S}^n$ is metrically homogeneous, given two points $x,y \in \mathbb{S}^n$ you must write down a formula for a metric isometry $f : \mathbb{S}^n \to \mathbb{S}^n$ such that $f(x)=y$. For proving that $\mathbb{B}$ is not metrically homogeneous, you must write down two specific points $x,y \in \mathbb{B}$, and then prove that there does not exist a metric isometry $f : \mathbb{S}^n \to \mathbb{S}^n$ such tyhat $f(x)=y$.